I worked with one of these almost unbranded 3.5" touchscreen, facing the same problems. So I had to work with an obsolete OS for a while, until I found a compatible drivers (device tree overlay to be precise) in Jessie.
DTOverlay route :
1 ) Check if your device is using such a dtoverlay, by looking in the /boot/config.txt on the modified OS (look for the line with "dtoverlay=somefilename").
2 ) Look on a fresh Rasbian if the "somefile.dtb" or "somefile.dtbo" exist in the /boot/overlays folder.
If it's not the case, you may try to copy this file from the modified OS to the new one.
In the worst case (i.e. raw copy not working), you may also have to find the source for this dtoverlay from the filename, in order to recompile it (not so hard, I even tweak mine to improve touchscreen FPS XD)
3 ) Copy the "dtoverlay=" line to rasbian /boot/config.txt file and reboot.
Kernel Module route :
Some touchscreen rely on specific kernel module to work. The one I know can be identified with :
lsmod | grep "fb_" && lsmod | grep "fbtft"